My main research interests are in nonlinear elliptic and parabolic partial differential equations and applications in geometry and optimal transportation. In particular, the regularity theory of Monge-Ampère equations, Hessian equations, and other variational problems.
I am also interested in related areas of geometry and physics, including geometry of convex bodies, minimal surfaces, surfaces of prescribed curvatures, and geometric flows.
Bayesian Sphere-on-Sphere Regression with Optimal Transport Maps, with T. L. J. Ng; K.-K. Kwong and A. Zammit-Mangion. Available at arXiv:2501.08492.
Optimal (partial) transport between non-convex polygonal domains, with S. Chen and Y. Li. Available at arXiv:2311.15655.
Global regularity in the Monge-Ampère obstacle problem, with S. Chen and X. Wang. Available at arXiv:2307.00262.
Boundary regularity for the second boundary-value problem of Monge-Ampère equations in dimension two, with S. Chen and X.-J. Wang. Available at arXiv:1806.09482.
Global $C^{1,\alpha}$ regularity for Monge-Amp\`ere equations on planar convex domains, with Q. Han and Y. Zhou. Mathematische Annalen, accepted.
Regularity of singular set in optimal transportation, with S. Chen. Inventiones Mathematicae, Vol. 242 (2025), 1--44.
Optimal transport problems with the squared distance cost function (in Chinese), with S. Chen. Scientia Sinica Mathematica, Vol. 54, no.5 (2024), 775--783.
$C^{2,\alpha}$ regularity of free boundaries in optimal transportation, with S. Chen and X.-J. Wang. Communications on Pure and Applied Mathematics, Vol. 77, no. 1 (2024), 731--794.
Regularity of optimal mapping between hypercubes, with S. Chen and X.-J. Wang. Advanced Nonlinear Studies, Vol. 23, no. 1, (2023) pp. 375--417.
Preface to the special issue: nonlinear PDEs and geometric analysis - dedicated to Neil Trudinger on the occasion of his 80th birthday, with J. Clutterbuck. Mathematics in Engineering, Vol. 5, no. 6, (2023), Paper No. 095, 5pp.
Non-uniqueness of solutions to the dual $L_p$-Minkowski problem, with Q.-R. Li and J. Lu. International Mathematics Research Notices, IMRN. (2022), no. 12, 9114--9150.
Global regularity for the Monge-Ampère equation with natural boundary condition, with S. Chen and X.-J. Wang. Annals of Mathematics, 194 (2021), 745--793.
A class of nonlinear optimisation and applications. Methods and Applications of Analysis, 28 (2021), 31--52.
Noncompact $L_p$-Minkowski problems, with Y. Huang. Indiana University Mathematics Journal, 70 (2021), 855--880.
Interior $C^2$ estimate for Monge-Ampère equation in dimension two. Proceedings of the American Mathematical Society, 149 (2021), 2479--2486.
Light refraction is nonlinear optimisation, with X.-J. Wang. Journal of Mathematical Study, 54 (2021), 142--163.
Optimal transport with discrete long range mean field interactions, with G. Loeper. Bulletin of Mathematical Sciences, 10 (2020), no. 2, 2050011, 26pp.
The $L_p$-Brunn-Minkowski inequality for $p<1$, with S. Chen; Y. Huang and Q.-R. Li. Advances in Mathematics, 368 (2020), 107166, 21 pp.
Stochastic continuity of random fields governed by a system of stochastic PDEs, with K. Du and F. Zhang. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 56 (2020), 1230--1250.
Global regularity of optimal mappings in non-convex domains, with S. Chen and X.-J. Wang. SCIENCE CHINA Mathematics, 62 (2019), 2057--2072.
Bergman-Toeplitz operators on fat Hartogs triangles, with T.-V. Khanh and P.-T. Thuc. Proceedings of the American Mathematical Society, 147 (2019), 327--338.
Bergman-Toeplitz operators on weakly pseudoconvex domains, with T.-V. Khanh and P.-T. Thuc. Mathematische Zeitschrift, 291 (2019), 591--607.
On the Cauchy problem for stochastic parabolic equations in Hölder spaces, with K. Du. Transaction of the American Mathematical Society, 371 (2019), 2643--2664.
Optimal transport with discrete mean field interaction, with G. Loeper. 2017 MATRIX Annals, Editors: David R. Wood, Jan de Gier, Cheryl E. Praeger, Terence Tao. MATRIX Book Series, Volume 2, Springer, pp.207--212, 2018.
Schauder estimates and application on stochastic partial differential equations. Surveys in Geometric Analysis 2017, Editors: Gang Tian, Qing Han, Zhenlei Zhang, Science Press, pp.94--115, 2018.
A degree theory for second order nonlinear elliptic operators with nonlinear oblique boundary conditions, with Y.Y. Li and L. Nguyen. Journal of Fixed Point Theory and Application, 19 (2017), 853--876.
A Schauder estimate for stochastic PDEs, with K. Du. C. R. Math. Acd. Sci. Paris, Ser. I, 354 (2016), 371--375.
On the classical solvability of near field reflector problems, with N.S. Trudinger. Discrete Contin. Dyn. Syst., 36 (2016), 895--916.
On the uniqueness of $L_p$-Minkowski problems: The constant $p$-curvature case in $\mathbb{R}^3$, with Y. Huang and L. Xu. Advances in Mathematics, 281 (2015), 906--927. Available at arXiv:1503.02358.
Boundary $C^{2,\alpha}$ estimates for Monge-Ampère type equations, with Y. Huang and F. Jiang. Advances in Mathematics, 281 (2015), 706--733.
On asymptotic behaviour and $W^{2,p}$ regularity of potentials in optimal transportation, with N.S. Trudinger and X.-J. Wang. Arch. Rat. Mech. Anal., 215 (2015), 867--905.
Interior a priori estimates for the Monge-Ampère equation, with X.-J. Wang. Surveys in Differential Geometry Volume 19, International Press (2014), 151--177.
Optimal transportation on the hemisphere, with S.-Y. A. Chang and P. Yang. Bulletin of the Institute of Mathematics, Academia Sinica 9 (2014), 25--44.
Degree theory for oblique boundary problems, (with Y. Y. Li and L. Nguyen). Oberwolfach Report 40/2013:26--29.
An obstacle problem for a class of fourth order equations, with B. Zhou. Journal of Differential Equations 254 (2013), 1306--1325.
Light reflection is nonlinear optimization. Calc. Var. Partial Differential Equations 46 (2013), 861--878.
Regularity of Monge-Ampère equations in optimal transportation. Bull. Austral. Math. Soc. 83 (2011), 173--176.
On Pogorelov estimates for Monge-Ampère type equations, with N.S. Trudinger. Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 1121--1135.
Interior $C^{2,\alpha}$ regularity for potential functions in optimal transportation, with N.S. Trudinger and X.-J. Wang. Comm. in Partial Differential Equations 35 (2010), 165--184.
Regularity in optimal transportation, (with N. S. Trudinger and X.-J. Wang). Oberwolfach Report 36/2009:17--20.
Hölder regularity of optimal mappings in optimal transportation. Calc. Var. Partial Differential Equations 34 (2009), no. 4, 435--451.
A multiagent-based domain transportation approach for optimal resource allocation in emergency management, with J. Zhang; M. Zhang and F. Ren. Multi-agent and Complex Systems, pages 19--32, Springer, Singapore, 2017.
A decentralised multi-agent system for emergency resource allocation in metropolitan regions, with J. Zhang; M. Zhang and F. Ren. Proceedings of the 15th International Conference on Autonomous Agents & Multiagent Systems (AAMAS 2016), pages 1482--1484, 2016.
Enable efficient resource deployment in multiple concurrent emergency events through a decentralised MAS, with J. Zhang; M. Zhang and F. Ren. Australasian Joint Conference on Artificial Intelligence, pages 56--68, 2016.
The singularity set of optimal transportation maps, with Z. Luo; W. Chen; N. Lei; Y. Guo; T. Zhao and X. Gu. Computational Mathematics and Mathematical Physics 62, 1313--1330, 2022.